- Home
- Standard 12
- Mathematics
For a polynomial $g ( x )$ with real coefficient, let $m _{ g }$ denote the number of distinct real roots of $g ( x )$. Suppose $S$ is the set of polynomials with real coefficient defined by
$S=\left\{\left(x^2-1\right)^2\left(a_0+a_1 x+a_2 x^2+a_3 x^3\right): a_0, a_1, a_2, a_3 \in R\right\} \text {. }$
For a polynomial $f$, let $f^{\prime}$ and $f^{\prime \prime}$ denote its first and second order derivatives, respectively. Then the minimum possible value of $\left(m_f+m_{f^{\prime}}\right)$, where $f \in S$, is. . . . . . . .
$5$
$8$
$9$
$10$
Solution
$f(x)=\left(x^2-1\right)^2 h(x) ; h(x)=a_0+a_1 x+a_2 x^2+a_3 x^3$
Now, $f(1)=f(-1)=0$
$\Rightarrow \quad f^{\prime}(\alpha)=0, \alpha \in(-1,1) \quad$ [Rolle's Theorem]
Also, $f^{\prime}(1)=f^{\prime}(-1)=0 \Rightarrow f^{\prime}( x )=0$ has atleast $3$ root, $-1, \alpha, 1$ with $-1<\alpha<1$
$\Rightarrow \quad f^{\prime \prime}( x )=0$ will have at leeast $2$ root, say $\beta, \gamma$ such that
$-1<\beta<\alpha<\gamma<1$
[Rolle's Theorem]
So, $\min \left( m _{f^{\prime \prime}}\right)=2$
and we find $\left( m _f+ m _{f^f}\right)=5$ for $f( x )=\left( x ^2-1\right)^2$.
Thus, Ans. $5$
